I kind of always took the, "that'll surprise the heck out of mom and dad" line as just a joke. I don't think he's actually referring to his parents.

More importantly...how does he afford a mortgage and other bills, including the cost of running all of the equipment in Darkwing Tower? When we first meet him, he's not involved in SHUSH. Later on, I'd accept that they pay him to freelance. But early on, how did he pay for his equipment and stuff? How did he pay for the parts to make the Ratcatcher? And the gas to run it?

I'd always assumed his parents are dead and left him an inheritance that he was very thrifty with.

I am the story that will read you real
Every memory that you hold dear...